Wave guide filters



' March 13, 1956 w. A. MILLER 2,738,468

WAVEGUIDE FILTERS Filed July 24, 1950 I 2 Sheets-Sheet 1 2& 1-9- We 0 504M! 0 t/ p o 0}/- kl, 04 La I 4 I 6 1 1 a 72m;

United States Patent M WAVE GUIDE FILTERS William A. Miller, Place,Y.-,' assigiio i" to Radio Corporation of America, a corporationofDelaware Application July 24, 1950, Serial No. 175,625

The terminal years of the term of the patent to be granted has beendisclaimed 1 Claim. (Cl. 333-73) This invention relates to microwaveguide filters and particularly to such filters consisting of a series ofwaveguide sections.

More particularly, the filter of this invention consists of a series ofwaveguide sections connected together in a predetermined sequence and atpredetermined angles, the guide sections having predetermined indexes ofrefractions and having predetermined lengths.

in my copending application Serial No. 166,350, filed June 6, 1950,and'in my copending application Serial No. 166,050, filed June 3, 1950,there were disclosed filters for a microwave system in whichinterference of other than the pass-band frequencies was accomplished byassembling together a series of waveguide sections having differentindexes of refraction. The waves entered and passed through thesefilters at zero angle of incidence, that is, the angle between thedirection of propagation and the normal to the junctions (hereinafterreferred to as surfaces) between waveguide sections, was zero degree.

In the embodiment of the present invention, the filter consists of aplurality of waveguide sections connected together and arrangedsymmetrically in both directions from the center guide section. Thefilter is, therefore, bidirectional. The outer or end guide sections areconnected to their contiguous intermediate guide sections at a criticalangle. This angle is determined such that there would be totalreflection of transmitted waves in the filter at the surfaces betweenthe outer guide section and its adjoining intermediate guide section,except for the predetermined index of refraction of the intermediateguide sections. In other words, total reflection of the transmittedwaves at the end intermediate surfaces is frustrated or prevented by theintermediate sections having predetermined indexes of refraction inrelation to the indexes of refraction of the outer guide sections forthe physical angle between the outer and intermediate guide sections.The index of refraction and the length of the center guide section issuch that, in combination with the outer and intermediate sections,interference of waves .of other than the band-pass frequencies isaccomplished.

The principal object of the invention is to provide a microwave filterin which a narrow pass-band is obtained by causing interference betweenwaves of other than the pass-band frequencies.

Another object of the invention is to provide a filter of highdiscrimination in which the transmitted waves are passed into onesection of the filter at a critical angle.

Another object of the invention is to provide a filter of highdiscrimination in which assembled sections of waveguides of criticaldimensions are connected together at critical angles.

Another object of the invention is to provide a filter of highdiscrimination in which the waves are fed into one section of the filterat an angle of incidence greater than zero degree.

Another object of the invention is to provide a filter of highdiscrimination in which waves within the filter would be totallyreflected except for a frustrating guide section of the filter.

Other objects and advantages of the invention will be apparent from thefollowing detailed description made with reference to the' accompanyingdrawings in which:

Fig. 1 is a simplified or line diagram of waveguides of the filter ofthis invention, showing the reflection of an' electromagnetic wave by asurface separating two waveguide sections having different indexes ofreflection;

Fig. 2 is a line diagram showing" the transmission of an electromagneticwave through three joined waveguide sections having different indexes ofrefraction;

Fig. 3 is a line diagram showing the transmission of an electromagneticwave through five waveguide sections having indexes of refraction of n2,n1, no, In, and m, respectively;

Fig. 4 is a line diagram showing the transmission and some of thereflections occurring when an electromagnetic wave enters and traversesfive waveguide sections having indexes of retraction of n2, 111, no, In,and nz, respectively;

Fig. 5 is a cross-section of one end of a frustrated total reflectionwaveguide filter;

Fig. 6 is a graph showing the relation of the index of refraction of awaveguide section to the ratio of (1) the wavelength of a wave in theguide to (2) the critical wavelength of the guide;

Fig. 7 is a top viewof a frustrated total reflection waveguide filter;

Fig. 8 is a front view of the frustrated total reflection filter in Fig.7;

Fig. 9 is a view in and 8; and

Fig. 10 is a graph showing the relation between the relative powertransmitted through the filter in Fig. 9 to the wavelength of the wavespropagated through the filter.

Referring to Fig. l, 0 and 1 are two waveguide sections having indexesof refraction of no and m, respectively, and separated by surface 0-1.Four (4) is the ray-tracer of an electromagnetic wave transmittedthrough section 1, striking surface 01 at point 5 and being reflected aswave 6. For this reflecting condition to exist, in must be greater thanno and the sine of the angle of incidence, 0 (the angle between thedirection of propagation and the normal 7 to the surface 01 at the pointof incidence 5) must be equal to or greater than the ratio of thesmaller to the larger index of refraction. As the index of refraction isdefined as the ratio of (l) the velocity of the electromagnetic Wave inair (taken as unity) to (2)' the group velocity of the wave in thewaveguide section, no equals 1/ V0 and n1 equals 1/ V1, where V0 and V1are the velocities of propagation of the wave in sections 0 and 1,respectively. For a reflecting condition at point 5, therefore, V0 mustbe greater than V1.

Referring to Fig. 2, waveguide section 0 is separated from section 1 bysurface 0-1 and section 2 is separated from section 1 by surface 2-1. Bydirecting the Wave 4 at surface 2-1 at an angle of incidence 02 and byintroducing between section 2 and section 0, respectively, a section 1of a proper length and index of refraction, wave 4 will enter section 1at' angle of refraction 61, be transmitted through section 1 at thevelocity V1, enter section 0 at an angle of refraction of B0 and betransmitted through section 0 at a velocity of V0. For such a condition,V1 must be greater than V2 and V2 must be greater than Vo.

It follows from the preceding that for other than the certain values ofangles of incidence, lengths and indexes of refraction of the waveguidesections total reflection of electromagnetic waves may be made to occuror, for

perspective of the filter in Figs. 7

certain values, as aforesaid, transmission through the Patented Mar. 13,19 56 3 sections may occur. The three sections, therefore, accomplishfrustrated total reflection.

To apply these conditions to provide a filter effective in eitherdirection of propagation through the filter, five sections areassembled, as in Fig. 3, in the order of indexes of refraction of 21,,n,, 11,, n,, and n respectively, with corresponding velocities ofpropagation of V2, V1, V0, V1, and V2, respectively. To constitute afrustrated total reflection filter, V must be less than V2 and V2 mustbe less than V1- In applying the principles of this invention to theconstruction of a filter, the angle of incidence 02 is made suflicientlylarge to ensure total reflection at the surface 21 if the index ofrefraction of section 1 is unity, that is, total reflection would occurin the filter at surface 21 if section 1 is a waveguide the index ofrefraction of which is unity. If n it, and n. are the indexes ofrefraction of waveguides relative to free space, then for totalreflection,

which is merely a statement of the law of total reflection, and, bySnells law of refraction,

sin 02 1 /n where "s is the index of refraction of the 5th sectionrelative to free space) and A is the wavelength of the radiation in freespace.

To design a filter for a particular waveguide system, then, it isnecessary, first to ensure an angle of incidence 02 such that totalreflection will occur at surface 21 when the index of refraction ofsection 1 is unity and then change the index of refraction of section 1by changing its dimensions to frustrate total reflection at this angleof incidence. As waveguides may be constructed over a wide range ofindexes of refraction, 02 may be determined by constructionconsiderations such as, for example, the mechanical arrangement ofsections in relation to sections of a waveguide system.

There are, however, limits that are imposed on the range of selection ofvalues of 02, which may be expressed as It is apparent that if the lefthalf of inequality (4) is not met, total reflection will not occur inthe absence of a frustrating section 1 and if the right half of theinequality (4) is not met, total reflection will occur at the surface2-1 even in the presence of the frustrating section 1. It is, therefore,concluded that l n n, (5

and since n must be greater than 11,, total reflection at the surface1-0 cannot occur, regardless of any real assumed value of 01 and 00 mustbe less than 01.

Frustrated total reflection filter In the filter arrangement in Fig. 3,the transmitted wave only is shown. There will, of course be reflectionsat the several surfaces, which phenomena make the arrangement respond asan interference filter. In the interests of simplicity, the reflectionsoccurring (Fig. 4) will be considered in only one direction oftransmission. As the reflections at the lower surface 2-1 and the lowersurface 1-0 are minor in strength and effect, they may be neglected.Likewise, the reflections transmitted through the lower section 2 may beneglected and only the reflections that throw back energy into section0, at 76 both the upper and lower surfaces 1-0, will be considered.

If R is the coefiicient of reflection and To is the coefiicient oftransmission at a surface between two sections, and zero absorption isassumed,

If 1r/J is the phase difference between (1) a wave transmitted fromsection 1 (lower) through medium 0 and into section 1 (upper) and (2)the same wave except reflected by upper surface 1-0, to the lowersurface 1-4), to and through the upper surface 10 and into section 1(upper), the intensity of the radiation (1) transmitted per unitincident radiation through such surfaces is given by the equation T, T,+4R site (11)) (7) Equation 8 (see Max Born, Optik, Springer 1933, 123),however, applies to parallel surfaces when the angle of incidence iszero. As the sections under consideration have different indexes ofrefraction, and the angle of incidence cannot be set equal to the angleof refraction, Equation 8 becomes where Rll-l is the coefiicient ofreflection at the lower or upper surface 0-1, 6, is the phase advance ona single passage of the wave through the 0 section and A is the phaseshift of the wave incident in the 0 section upon reflection from one ofsurfaces 01.

The phase advance 6,, on a single passage of a wave through the sectionmay be defined by the equation 5,=21rn, cos 0, (10) where d5 is theperpendicular equivalent length of the 5th section.

It will be noted from Equation 9 that the value of (I) is a maximum whenwhere m is an integer. The values of A and Ro-1 may, therefore, becalculated as follows:

Reflection coefficient (R) for a multi-section filter The relationbetween index of refraction n and the angle of incidence 0 for aparticular (5th) refracting section (ts) may be defined as it, cos 0,t,'

K, cos 0.-1 (13) Where its is defined in Equation 3.

The effective index ofrefraction of each of the s layers (Ts) would,therefore, be

I 1 --1 T,+1 tan 5. (14) The reflection coefiicient of a multi-sectionfilter may then be found by determining from the known values of t +1and fi found from Equations 13 and 10, respectively. The values of T3for successively decreasing subscripts are then derived until T1 isfound. From this value, the value of R may be calculated, using Equation12.

To evaluate R for a multi-section filter in which the electric vector isparallel to the plane of incidence, Equation 13 is written as lc, cos 0(16) and the values obtained are substituted in Equations 14 and 15 toobtain the value of R under these conditions. It will be noted thatEquation 16 is not the reciprocal of Equation 13.

Obtaining the value of R for a frustrated total reflection filter Thefirst step in evaluating the reflection coeflicient R for a frustratedtotal reflection filter is to determine the coefficient of reflection atthe boundary surfaces of the interfering waveguide 0. It is apparentthat the value of R at only one boundary surface and for only onedirection of propagation need be obtained.

In Fig. 5, which is the waveguide equivalent of the line diagram Fig. 2,wave 4 is transmitted into the waveguide 8 at an angle 00 and passesthrough frustrating waveguide section 9 of index of refraction 11,, thebound ary surface between the two guide sections being indicated by line10. After traversing guide section 9, the wave is transmitted into guidesection 11, which is connected to guide section 9 at an angle of 92degrees.

From Equation 14, the value of T1 for the surface 2-1 is and the valueof 61 in Equation 10 become imaginary and may be defined as (19a) 19b)(190) T1 then becomes T z't (-il +tanh 'y) 1 l-it tanh 'y The value ofT1 may be substituted in Equation 12 to obtain the value of R0-1.

Evaluation of A In deriving the value of A, it may be assumed that totalreflection occurs. The error introduced by this assumption is veryslight for a narrow band filter such as this invention provides. Theassumption may be stated as tan A=l (21) or T1=it,' and From Equation 22and by the laws involved in phase change upon reflection,

Calculation of the dimensions of a frustrated total reflection filter Inthe practical application of the principles of this invention tofilters, certain values and conditions are set and determined by thewaveguide system into which the filter is to be included. For example,let the wavelength to be transmitted in the guide system be 1 cm. andthe radiation be of the TEM mode. The index of refraction of waveguidesmay be determined from the graph in Fig. 6, for selected ratios of (1)the wavelength in the guide to (2) the cutoff wavelength. Hence, thedimensions of the waveguide may be selected for a mean value of n,, forexample, n =1.5.

From Equation 1, with a value of n,=l.5, 62 is greater or equal to 42degrees. Selecting a reasonable value of 11 for the frustrating guidesection of 11 equals 1.25, from Equation 4, 02 is equal to or less than56 degrees. The value of 02 equals 50 degrees is thus found to besatisfactory.

The value of the index of refraction n of section 0 must be somewhatgreater than n but, from a consideration of the trend of the curve inFig. 2, a value of n =2.50 is selected to avoid high dispersion in thissection.

The value of k is determined by For 0 =50, sin 0,=0.766, and cos0,=0.643.

From Equation 2,

sin 62 and cos 0 =0.886 From Equation 10 and setting the length of the 0section equal to 1.11

6 =4.9551r and from Equation 11 4.9951r+A=51r and A=0.0451r FromEquation 23 t,'=-tan 4 or From the Inequality (4), the radiationpropagated in the direction of the arrow in Fig. 5, the right part ofInequality (4) must be used for the angle of incidence in thefrustrating guide section, and

sin 0 7 cos 0,, therefore equals:

cos 0,- =i(1.04l) =0.2i From Equation 10 t, is obtained from Equation 13Applying the determined values to Equation 20 Substituting this value ofT1 in Equation 12,

and R0-1 =0.895.

From the data hereinbefore determined, the dimensions of the filter andits performance may be calculated. The dimensions of waveguide sectionsin Fig. 7 (a top view of the filter) and in Fig. 8 (a side elevation ofthe filter) corresponding to the reference characters are stated orcalculated as follows:

From Fig. 6, with 110:2.50, M/ \=0.92 and as 101:1 cm. and \c=1.09 cm.,Wo, therefore, equals 0.545 cm. Similarly for values n and n w equals0.78 cm. and w, equals 0.87 cm.

d, and d were calculated on the basis of their radiation length and,therefore, their linear length is found by multiplying the values of d,and d by their respective indexes of refraction.

d,'=0.605.1.25=0.756 cm. and

d =1.11.250=2.79 cm.

Internal resistors 12 are provided in the frustrating section 1 todampen any spurious cavity modes. The

length l of the filter is not critical, if it is several waveofrefraction. As filters in microwave systems are energized by radiationsof only slightly wider band of frequencies than the band-widths of thefilters, i, may be assumed to be a constant, that is, the waveguidedispersion may be neglected. On the other hand, 6 =k/ which is not aconstant.

The relative power transmitted in relation to the Wavelength in ems. hasbeen plotted in Fig. 9 using the equation 1 1+325 sin (1.76+13.9/)\) Fora pass-band at 1 cm., the pass-bands closest to the 1.0 cm. pass-band,as calculated from Equation 21, are located at 1:082 cm. and \=1.3 cm. Afilter such as disclosed in my pending application Serial No. 166,050,filed June 3, 1950, will provide more than ample isolation for thepass-band centered at 1 cm.

There has thus been disclosed herein a bidirectional, interference typeof waveguide filter consisting of waveguide sections in series ofpredetermined indexes of refraction and lengths, the end sections ofwhich are connected to intermediate sections at predetermined angles toproduce at the joint surfaces of these sections, frustrated totalreflection and accompanied interference filtering.

What is claimed is:

A microwave guide filter for selectively transmitting microwave energyat a predetermined frequency comprising, a first and central waveguidesection having a longitudinal axis and an index of refraction of n atsaid frequency, second and third waveguide sections connected toopposite ends of said first waveguide section each having indexes ofrefraction 11 at said frequency and their longitudinal axes parallel tothe axis of said first waveguide section, means within said second andthird waveguide sections for damping spurious modes within said secondand third waveguide sections, and fourth and fifth waveguide sectionsconnected to the outermost ends of said second and third waveguidesections each of said fourth and fifth waveguide sections having indexesof refraction n at said frequency, said fourth and fifth waveguidesections being connected to said second and third waveguide sections,respectively, with the longitudinal axes of said fourth and fifthwaveguide sections inclined with respect to the longitudinal axes ofsaid second and third waveguide sections at an angle 0,. where s sin 0,5"2 "2 References Cited in the file of this patent UNITED STATES PATENTS2,129,712 Southworth Sept. 13, 1938 2,142,138 Llewellyn Jan. 3, 19392,261,130 Applegate Nov. 4,1941 2,407,911 Tonks Sept. 17,1946 2,436,828Ring Mar. 2, 1948 2,531,437 Johnson Nov. 28, 1950 2,576,186 Malter Nov.27, 1951 2,601,806 Turner July 1, 1952 2,623,121 Loveridge Dec. 23, 1952

